Which self homeomorphisms preserve measure on a torus, apart from affines? Affine is the composition of rotation and automorphism. Measure is the Lebesgue measure.
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2$\begingroup$ @YemonChoi I believe the flow generated by any solenoidal vector field should preserve measure, so there should be many examples (unlike the case in one-dimension). $\endgroup$– Gabe KCommented Dec 2 at 1:53
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1$\begingroup$ What do you mean by a "continue automorphism"? This does not parse even on the syntax level. It is a general fact that the group of measure preserving diffeomorphisms is infinite-dimensional for every nonempty manifold of dimension $>1$ and any smooth measure. You can read Banyaga's book to learn more about such groups. $\endgroup$– Moishe KohanCommented Dec 2 at 1:58
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1$\begingroup$ @MoisheKohan "continue" could be French and mean continuous (then it makes sense on the syntax level) $\endgroup$– Ville SaloCommented Dec 2 at 2:32
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1$\begingroup$ @VilleSalo: Maybe, but a "continuous automorphism" is very unclear mathematically speaking. $\endgroup$– Moishe KohanCommented Dec 2 at 2:36
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1$\begingroup$ The word "automorphism" by itself is meaningless. You have to specify the structure which your map preserves. For instance, I can regard the torus as an abstract abelian group and talk about automorphisms of this group. Or I can regard the torus as a complex manifold, then automorphisms will be biholomorphic self-maps. Or I can regard the torus as a topological manifold, then automorphisms will be self-homeomorphisms. $\endgroup$– Moishe KohanCommented Dec 2 at 2:46
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2 Answers
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As a simple example, if you take any divergence-free vector field, the flow generated by this vector field is incompressible, so preserves measures.
In fact, there are a huge number of measure preserving diffeomorphisms of the torus, so many that they cannot be classified.
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It is very easy to construct many measure-preserving homeomorphisms which are piecewise affine. I illustrate this with an example (easily modifiable and generalizable ad nauseam) involving the standard torus $\mathbb R^2/\mathbb Z^2$: Consider the piecewise linear map defined by $(x,y)\longmapsto (x,y)$ for $0\leq x\leq 1$ and $0\leq y\leq 1/2$ and by $(x,y)\longmapsto (x+2y-1,y)$ for $0\leq x\leq 1$ and $1/2\leq y\leq 1$.
answered Dec 2 at 8:13